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Multiple alignment points in LaTeX

If I have two strings of equalities within the same align or alignat environment, and the first one is long enough to need two lines, I would want the result to look like
Code:
A+B+C=D=E
=F=G=H
I=J
rather than
Code:
A+B+C=D=E
=F=G=H
I=J
Is there a way to do this? I suppose it can be done with an array, but I thought either align or alignat would be able to handle this. I get garbage results like these:

\begin{alignat}{3}
&F_k &=(f\chi_E)^{-1}(v_k)=\{x\in X|f(x)\chi_E(x)=v_k\}=\{x\in X|f(x)=v_k\}\cap E\\
&&=f^{-1}(v_k)\cap E=E_k\cap E\in\Sigma,\\
&\mu\big(F_k)=\mu(E_k\cap E)\leq\mu(E_k)<\infty.
\end{alignat}

\begin{alignat}{4}
&F_k && =(f\chi_E)^{-1}(v_k) =\{x\in X|f(x)\chi_E(x)=v_k\}=\{x\in X|f(x)=v_k\}\cap E\\
&&&=f^{-1}(v_k)\cap E=E_k\cap E\in\Sigma,\\
&\mu\big(F_k)=\mu(E_k\cap E)\leq\mu(E_k)<\infty.
\end{alignat}

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